2015 AMC 10A Problems/Problem 15
Consider the set of all fractions , where and are relatively prime positive integers. How many of these fractions have the property that if both numerator and denominator are increased by , the value of the fraction is increased by ?
You can create the equation
Cross multiplying and combining like terms gives .
This can be factored into .
and must be positive, so and , so and .
This leaves the factor pairs: and
But we can't stop here because and must be relatively prime.
gives and . and are not relatively prime, so this doesn't work.
gives and . This doesn't work.
gives and . This does work.
We found one valid solution so the answer is .